Lemma 6.14.101088A4C2B5CA

Proof. (1) By proposition 5.2.6 there is a smooth partition of unity $(\chi _ a)$ subordinate to the cover $(V_ i)$; for each $a$ choose an index $k(a)$ with $\operatorname{Supp}\chi _ a\subset V_{k(a)}$, and put $\rho _ k=\sum _{k(a)=k}\chi _ a$. Since the family $(\chi _ a)$ is locally finite, each $\rho _ k$ is smooth, the family $(\rho _ k)$ is locally finite, $\sum _ k\rho _ k=1$, and $\operatorname{Supp}\rho _ k\subset V_ k$, because a locally finite union of closed sets is closed. For each $i$ put

where the $k$th term, defined on $V_ i\cap V_ k$, is extended by zero to the rest of $V_ i$. Each term is smooth on $V_ i$, since it is smooth on $V_ i\cap V_ k$ and vanishes on the open set $V_ i\setminus \operatorname{Supp}\rho _ k$, and these two open sets cover $V_ i$. By local finiteness, $|e_ i|^2$ is smooth on $V_ i$, and it is positive, because the $\rho _ k$ are nonnegative with sum one and every $g_{ik}$ is nowhere zero. On $V_ i\cap V_ j$, the cocycle relation $g_{ik}=g_{ij}g_{jk}$ holds on $V_ i\cap V_ j\cap V_ k$, hence wherever $\rho _ k\neq 0$, so

Thus these squared norms form a smooth quadratic Hermitian metric on $L$.

(2) On $V_ i\cap V_ j$ we have $\log |e_ i|^2-\log |e_ j|^2=\log |g_{ij}|^2$, which is pluriharmonic, since $g_{ij}$ is holomorphic and nowhere zero (section 5.4). Hence the forms $-\mathrm{dd}^{\mathrm{c}}\log |e_ i|^2$ on $V_ i$ agree on the overlaps and define a $(1,1)$-form $\eta $ on $X$. It is smooth, because each $\log |e_ i|^2$ is a smooth real function. Moreover $\eta =-\mathrm d(\mathrm d^{\mathrm c}\log |e_ i|^2)$ on $V_ i$, by (5.2.10), and $\mathrm d^{\mathrm c}$ maps real forms to real forms; hence $\eta $ is real, and it is closed because $\mathrm d^2=0$.

(3) The functions $|e_ i|_{h_0}^2\mathrm e^{-f}$ are smooth and positive on $V_ i$, and they satisfy the compatibility condition of definition 6.14.9 because the functions $|e_ i|_{h_0}^2$ do. The curvature of $h_0\mathrm e^{-f}$ is $-\mathrm{dd}^{\mathrm{c}}\log \bigl (|e_ i|_{h_0}^2\mathrm e^{-f}\bigr )=-\mathrm{dd}^{\mathrm{c}}\log |e_ i|_{h_0}^2+\mathrm{dd}^{\mathrm{c}}f=\eta _0+\mathrm{dd}^{\mathrm{c}}f$ on $V_ i$. Conversely, let $h_1$ be a smooth quadratic Hermitian metric on $L$. On $V_ i$ the function $|e_ i|_{h_1}^2/|e_ i|_{h_0}^2$ is smooth and positive, and on $V_ i\cap V_ j$ it equals $|e_ j|_{h_1}^2/|e_ j|_{h_0}^2$, since both numerator and denominator are multiplied by $|g_{ij}|^2$. Hence these quotients define a smooth positive function on $X$, which we write as $\mathrm e^{-f}$ with $f$ smooth and real; then $h_1=h_0\mathrm e^{-f}$. The function $f$ is unique, since $\mathrm e^{-f}$ must equal these quotients. $\square$